where \(dn(z,m)\)  and \(cn(z,m)\)  are also Jacobi elliptic functions [1]}, and \(\eta \)  may be complex. The integral \(\int _{0}^{\eta }a(\eta ^{\prime })d\eta ^{\prime }\)  depends on how the contour from \(\eta ^{\prime }=0\)  to \(\eta ^{\prime }=\eta \)  winds around the poles of \(a(\eta ^{\prime })\) . This in turn corresponds to the fact that the answer (REF ) depends
on \({\rm ln}(z)\) , which is only defined modulo \(2\pi i\) , so that \(t\)  itself is only well defined modulo \(\Delta t=2\pi i(\alpha /\beta )m^{-1/2}=2\pi i\sqrt{3/\lambda }\) . But a system whose time coordinate is only defined modulo an imaginary shift \(t\rightarrow t+\Delta t\)  (or, in other words, whose \(t\)  is curled into a circle of circumference \(\Delta t\)  in the imaginary time direction) corresponds to a system at finite temperature \(T=i/\Delta t\) . In this way, we precisely recover the well-known expression for the de Sitter temperature
\(T_{dS}=H/2\pi \) 
